When you use the expression ‘At the Drop of a Hat’ you mean that something is
happening instantly, without any delay.
Example of use: “We’re all packed and ready to go; we can leave at the drop of a hat.”
The phrase ‘at the drop of a hat’ originates in the 19th century. During that time it was common to signal the beginning of a fight or race by either dropping a hat or sweeping it in a rapid downward motion.